reserve Y for TopStruct;
reserve X for non empty TopSpace;
reserve X for almost_discrete non empty TopSpace;

theorem Th51:
  for A being Subset of X holds (for x being Point of X st x in A
  ex F being Subset of X st F is closed & A /\ F = {x}) implies A is discrete
proof
  let A be Subset of X;
  assume
A1: for x being Point of X st x in A ex F being Subset of X st F is
  closed & A /\ F = {x};
  now
    let x be Point of X;
    assume
A2: x in A;
    now
      consider F being Subset of X such that
A3:   F is closed and
A4:   A /\ F = {x} by A1,A2;
      take F;
      thus F is open by A3,TDLAT_3:22;
      thus A /\ F = {x} by A4;
    end;
    hence ex G being Subset of X st G is open & A /\ G = {x};
  end;
  hence thesis by Th31;
end;
